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The Scientific World Journal
Volume 2014 (2014), Article ID 514854, 14 pages
http://dx.doi.org/10.1155/2014/514854
Research Article

Periodic Solutions for Nonlinear Integro-Differential Systems with Piecewise Constant Argument

Departamento de Matemática, Facultad de Ciencias Básicas, Universidad Metropolitana de Ciencias de la Educación, Santiago, Chile

Received 31 August 2013; Accepted 10 October 2013; Published 12 January 2014

Academic Editors: R. Adams, A. Ibeas, and M. Inc

Copyright © 2014 Kuo-Shou Chiu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the existence of the periodic solutions of a nonlinear integro-differential system with piecewise alternately advanced and retarded argument of generalized type, in short DEPCAG; that is, the argument is a general step function. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such equations are obtained. In the process we use Green’s function for periodic solutions and convert the given DEPCAG into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii’s fixed point theorem to show the existence of a periodic solution of this type of nonlinear differential equations. We also use the contraction mapping principle to show the existence of a unique periodic solution. Appropriate examples are given to show the feasibility of our results.